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Learning Spatiotemporal Tubes for Temporal Reach-Avoid-Stay Tasks using Physics-Informed Neural Networks

Ahan Basu, Ratnangshu Das, Pushpak Jagtap

TL;DR

The paper addresses safety-critical, time-bounded reach-avoid-stay tasks for uncertain control-affine MIMO systems by learning Spatiotemporal Tubes (STT) with a Physics-Informed Neural Network (PINN). It introduces PINSTT to jointly learn the STT center and radius from collocation data and proves formal guarantees via a Lipschitz-based validity condition, enabling an approximation-free closed-form controller that keeps system outputs inside the STT. The approach is verified on an omnidirectional robot and a quadrotor navigating dynamic and static obstacles, demonstrating both offline STT training efficiency and fast online control. This framework offers a scalable, model-free route to safety guarantees in time-critical navigation tasks with unknown dynamics and disturbances.

Abstract

This paper presents a Spatiotemporal Tube (STT)-based control framework for general control-affine MIMO nonlinear pure-feedback systems with unknown dynamics to satisfy prescribed time reach-avoid-stay tasks under external disturbances. The STT is defined as a time-varying ball, whose center and radius are jointly approximated by a Physics-Informed Neural Network (PINN). The constraints governing the STT are first formulated as loss functions of the PINN, and a training algorithm is proposed to minimize the overall violation. The PINN being trained on certain collocation points, we propose a Lipschitz-based validity condition to formally verify that the learned PINN satisfies the conditions over the continuous time horizon. Building on the learned STT representation, an approximation-free closed-form controller is defined to guarantee satisfaction of the T-RAS specification. Finally, the effectiveness and scalability of the framework are validated through two case studies involving a mobile robot and an aerial vehicle navigating through cluttered environments.

Learning Spatiotemporal Tubes for Temporal Reach-Avoid-Stay Tasks using Physics-Informed Neural Networks

TL;DR

The paper addresses safety-critical, time-bounded reach-avoid-stay tasks for uncertain control-affine MIMO systems by learning Spatiotemporal Tubes (STT) with a Physics-Informed Neural Network (PINN). It introduces PINSTT to jointly learn the STT center and radius from collocation data and proves formal guarantees via a Lipschitz-based validity condition, enabling an approximation-free closed-form controller that keeps system outputs inside the STT. The approach is verified on an omnidirectional robot and a quadrotor navigating dynamic and static obstacles, demonstrating both offline STT training efficiency and fast online control. This framework offers a scalable, model-free route to safety guarantees in time-critical navigation tasks with unknown dynamics and disturbances.

Abstract

This paper presents a Spatiotemporal Tube (STT)-based control framework for general control-affine MIMO nonlinear pure-feedback systems with unknown dynamics to satisfy prescribed time reach-avoid-stay tasks under external disturbances. The STT is defined as a time-varying ball, whose center and radius are jointly approximated by a Physics-Informed Neural Network (PINN). The constraints governing the STT are first formulated as loss functions of the PINN, and a training algorithm is proposed to minimize the overall violation. The PINN being trained on certain collocation points, we propose a Lipschitz-based validity condition to formally verify that the learned PINN satisfies the conditions over the continuous time horizon. Building on the learned STT representation, an approximation-free closed-form controller is defined to guarantee satisfaction of the T-RAS specification. Finally, the effectiveness and scalability of the framework are validated through two case studies involving a mobile robot and an aerial vehicle navigating through cluttered environments.

Paper Structure

This paper contains 16 sections, 4 theorems, 24 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notation
  4. System Definition
  5. Problem Formulation
  6. Physics-informed Neural Spatiotemporal Tube (PINSTT)
  7. Proof:
  8. Formal verification
  9. Proof:
  10. Formulation of loss functions and training procedure
  11. Controller Synthesis
  12. Proof:
  13. Case Study
  14. Omnidirectional Robot
  15. Quadrotor
  16. ...and 1 more sections

Key Result

Lemma 3.1

The time-varying ball $\Gamma(t)$ characterized by center ${\mathbf{c}}(t)$ and radius ${\mathbf{r}}(t)$, as mentioned in Definition def:stt, is considered a valid STT satisfying the T-RAS specification in Definition def:tras if the following conditions are satisfied with $\eta \leq 0$: where ${\mathbb B}({\mathbf{c}}_{\mathbf{S}}, {\mathbf{r}}_{\mathbf{S}}) \subseteq {\mathbf{S}}$, ${\mathbb B}(

Figures (2)

  • Figure 1: The constructed STT and the omnibot trajectory in a dynamic environment with time-varying obstacles.
  • Figure 2: The drone trajectory in a static environment with static obstacles.

Theorems & Definitions (10)

  • Definition 2.1: Temporal Reach-Avoid-Stay Task
  • Definition 2.3: Spatiotemporal Tubes for T-RAS tasks
  • Remark 2.4
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 3.6
  • Theorem 4.1